On an indefinite semilinear elliptic problem on \(\mathbb R^N\) (Q1883086)

This paper is devoted to the study of non-negative weak solutions of the bifurcation elliptic problem \(-\Delta u=\lambda h(x)u+g(x)u^p\) in \({\mathbb R}^N\), where \(N>2\), \(\lambda\in {\mathbb R}\), \(1<p<(N+2)/(N-2)\), and \(h(x)\) and \(g(x)\) are sign changing potentials. The main result of the present paper establishes the existence of a continuum of positive solutions, bifurcating from the principal eigenvalues \(\lambda_{1,h}\) and \(-\lambda_{1,-h}\) of the associated linear problem. The proof is based on the global bifurcation theorem of \textit{P. Rabinowitz} [J. Funct. Anal. 7, 487--513 (1971; Zbl 0212.16504)], combined with standard \textit{a priori} estimates of positive solutions.